DESIGNING A FEEDBACK VORTICITY CONTROL SYSTEM FOR THE FLOW PAST A CIRCULAR CYLINDER

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1 Proceedings of the IASTED International Conference Modelling, Identification and Control (MIC 3) February - 3, 3 Innsbruck, Austria DESIGNING A FEEDBACK VORTICITY CONTROL SYSTEM FOR THE FLOW PAST A CIRCULAR CYLINDER H. Deniz Karaca, G. Deniz Özen, Cosku Kasnakoglu Department of Electrical Engineering, TOBB University of Economics and Technology, Ankara, Turkey Deparment of Physics, Middle East Technical University, Ankara, Turkey ABSTRACT In this work a novel approach for the feedback control of vorticity behind an immersed circular cylinder is considered. The technique is based on determining the measurement points behind the cylinder for vorticity magnitude values and injection points on the cylinder for the control input, applying a significantly exciting input actuation, collecting measurement data to estimate the dynamical model of the flow field using system identification and finally designing a feedback control system using the dynamical model obtained. Numerical simulation results show that the designed feedback control system can reduce the vorticity magnitude values to a desired reference level. KEY WORDS Flow control, vortex suppression, circular cylinder, dynamical modelling, system identification, internal model control. Introduction Analyzing the fluid flow around bluff bodies is one of the important problems in fluid mechanics. Bluff bodies attract the attention of researchers due to the unsteady wake region formed behind the body as a result of fluid flow. Circular cylinder geometry is one of the popular cases of bluff bodies since it represents key characteristics of complex flow such as unsteady wake flow, powerful flow separation, turbulent behavior at low Reynolds numbers and vortex shedding. For these reasons circular cylinders have been a major interest of studies in the field for nearly a century [, ]. Due to its simple geometry and typical behavior of flow separation, study of viscous flows around the circular cylinder has recently become a standard testbed for modeling, analysis and control. Also, understanding the behavior of flow around a cylinder is useful for modeling and controlling more complex flows [3]. Analyzing and controlling flow past a circular cylinder has significant importance on both theoretical and practical areas []. Hence, a variety of methods have been proposed to control such flows, e.g. including a moving surface, accelerating the boundary layer by blowing, sucking and injecting gas, preventing transition within the boundary layer using appropriate shaped objects, surface cooling and so on []. Drag reduction is a major problem discussed under the topic of flow past a cylinder and strategies towards this goal were investigated both theoretically and experimentally. Efforts in this direction include the work by He et. al. [] who examined computational methods for active control and drag optimization of incompressible viscous flow past a cylinder, Pastoor et. al. [7] who investigated strategies for an elongated D-shaped body and Bergmann et. al. [] who studied the optimal control approach for the active control and drag optimization of incompressible viscous flow past a circular cylinder. Another main goal in the investigation of fluid flow past a cylinder is to obtain strategies to control the vortex shedding. Examples include Fan [9] who examined whether vortex shedding can be controlled by plasma actuators using an immersed boundary method with the combination of an empirical plasma model. Protas [] used cylinder rotation as the means for actuation, and based on a linearized Föppl system, he showed that the problem is stabilizable, but not controllable. To describe the flow past a cylinder mathematically one uses the Navier-Stokes partial differential equations (PDEs). Although these equations are very accurate, they are quite complex and usually very hard (if not impossible) to solve analytically. For this reason one usually employs computational fluid dynamics (CFD) simulations to obtain numerical solutions, and myriad studies for various flow configurations have been reported using this approach, e.g. Park et. al. [] carried out a computational study of the feedback control of von Karman vortex shedding behind a circular cylinder at low Reynolds numbers. A common approach to simplifying Navier-Stokes PDEs is to apply reduced-order modeling techniques such as Proper Orthogonal Decomposition (POD) and Galerkin Projection (GP) so as to obtain a simpler representation of the flow field. Examples of works using reduced order models include Tadmor et. al. [] who proposed a method to include dynamic mean field representations in low order Galerkin models and Noack et. al. [-3] who obtained Galerkin approximations achieving accurate representations of the unstable solution for the cylinder wake by adding shift-modes and proposed a system reduction strategy for spectral and Galerkin models. Despite being simpler than the Navier-Stokes equations, the DOI:.3/P

2 models produced by POD/Galerkin approaches are still nonlinear, and hence challenging to work on directly [,, ]. Moreover the computations required for the simplifications are quite involved and prone to errors [7, ]. In this study we consider a modeling and control method for fluid flow around a circular cylinder which does not rely on POD/GP, but instead directly uses the measurement data saved from CFD simulations. These data are processed using system identification (SI) to produce linear models of the flow system, with the inputs and outputs determined according to the specifics of the flow problems so as to keep model complexity to a minimum. The model s capability in reconstructing the actual flow dynamics is demonstrated through the verification data and a linear controller design is constructed for the model using automated control synthesis techniques. The controller s success in suppressing the vortex street past the cylinder is demonstrated via CFD simulations..problem Description The flow over a cylinder becomes turbulent at relatively low Reynolds numbers (about 7) and causes an unsteady separation of flow. As a result, a repeating pattern of swirling vortices are formed called the von Karman vortex street. The problem geometry studied in the paper is illustrated in Figure, together with the mesh structure utilized to solve the underlying Navier-Stokes PDEs governing the flow. The Navier-Stokes PDEs are solved using the Navierd, a CFD tool for MATLAB [9]. We have significantly modified and extended to program to accommodate our needs for the study in this paper and these modifications will be described in detail in the upcoming sections. Nodes, 7 Triangles diameter centered at the origin,. A fluid of kinematic viscosity.7 / flows into the flow domain from the left boundary at a velocity of /, i.e. are the streamwise and transverse components of the flow velocity respectively. The top and bottom boundaries are assumed free-slip, i.e. and on these boundaries, where is the normal direction to the boundary. The fluid is assumed to exit the flow domain at constant pressure from the right boundary. The cylinder is assigned no-slip boundary conditions, i.e. on the cylinder surface. With these settings the Reynolds number for the flow can be computed to be approximately and since 7 we would expect to see a repeating pattern of vortex shedding, which is confirmed from running the CFD simulations for about seconds (Figure ). Observing the figure one can clearly see the vortex shedding pattern. The goals of this study are: ) Obtain a simple and linear dynamical model to represent this behavior and ) Design a means to actuate the flow so as to attenuate the vortex shedding. - Figure. The velocity field resulting from running the CFD simulation for about seconds Methodology 3. CFD Simulations for Generating Input-Output Data Figure. Problem geometry and the mesh used in simulations The first step is to perform CFD simulations of the flow using Navierd to collect data to be used later in system identification. For this purpose we need to determine the input and output for the problem. For the actuation input we select two small regions at the top and bottom of the cylinder and assume that we can blow/suck air from these locations. Since our goal is to suppress the vortex shedding past the cylinder, we select a group nodes behind the cylinder from where we take vorticity measurements. The vorticity can be computed as The boundary conditions are parameters used for the CFD simulations are as follows: The flow domain is Ω,,, with a cylinder of 3

3 where and are the streamwise and transverse components of flow velocity. As the output of the system we take the mean vorticity magnitude over the group of nodes, i.e. y.. Input and output signals where denotes the vorticity at the th measurement node and is the total number of nodes selected for measurement. Normally, CFD tool Navierd does not have an interface for selecting input/output regions, but we have modified the program to allow for such a selection. The actuation input and the measurement region selected through the program are shown in Figure 3. The actuation holes can be seen as tiny magenta points on the cylinder and the measurement region is the magenta rectangular area towards right of the cylinder. u Time Figure. Chirp signal input ( ) and the resulting measured output ( ) 3. System Identification for Obtaining the Model Once the input-output data is obtained from CFD simulations, the goal is to find a dynamical that fits the data. Many model types have been experimented with, but the best results were obtained when a process model of the following type e 3 Figure 3. The Selected Actuation Input and Measurement Points To excite the system sufficiently and reveal enough underlying dynamics it is common practice in system identification to apply a signal that contains a variety of frequencies. For this purpose we used a chirp signal of unit magnitude, duration of, where the frequency varies from. to for the first and then goes from back to. for the next. The input applied and the output resulting from CFD simulations using Navierd are shown in Figure. In the figure one can see that the input and output signals are partitioned into two parts shown in green and red. The green part is the first of the data and will be used for building the dynamical model using system identification. The red part is the last of the data, which is reserved for validating the results of system identification. The next step is to perform system identification on this inputoutput data to obtain the dynamical model. was used, which describes the system dynamics in terms of static gain K, damping coefficient, inverse natural frequency process zero T, time delay T, time constant T and an integrator. These parameters were determined using MATLAB System Identification Toolbox where the iterative prediction-error minimization (PEM) technique with an adaptive version of subspace Gauss-Newton approach was employed []. The resulting values for the parameters are shown in Table Table. Parameter values for the process model obtained from system identification.

4 These parameter values were obtained using the first portion of the data from CFD simulations (see Figure ). To verify the model, we use the second portion of this data, i.e. we apply the input for the second portion to the model obtained from system identification and compare the results with the measured output obtained from CFD (Figure ). It can be seen from the figure that the measured and simulated outputs are quite close and therefore it can be concluded that the model represents the flow dynamics with reasonable accuracy... Measured and simulated model output The input to the controller is the error where is the system output (mean vorticiy magnitude) and is the reference value (which will normally be zero since we would like to attenuate the vorticity as much as possible). The controller output, which is the input to be applied to the system (velocity of the stream injected to the flow domain through the holes on the cylinder). The final locations of the closed-loop poles can be seen in the root locus plot (Figure ). Note that all poles are in the left-hand plane so the closed loop system is stable, but a pair of complex poles are very close to the imaginary axis resulting in a slow response with large overshoot. This is confirmed by the step response of the closed-loop system shown in Figure 7.. Root Locus Simulated output Measured output Time Figure. Measured and simulated model outputs Imag Axis Controller Design The current step is to design a controller for the model obtained in the previous section to regulate the system output, which is the mean vorticity magnitude. Since the model obtained is linear, numerous of standard and automated design methods exist for obtaining the controller such as proportional integral derivative (PID) tuning techniques, internal model control (IMC) design methods, linear quadratic Gaussian (LQG) synthesis and optimization-based design. For the current problem various compensators of different orders were constructed using these methods with the help of MATLAB Control Systems Toolbox. The best results were obtained using IMC tuning methods [,] followed by some custom editing on the root locus plot, which resulted in the following controller Real Axis Figure. Root locus plot and closed-loop pole locations Amplitude Step Response Time (sec) Figure 7. Step response of the closed-loop system where Results In order to observe the effect of the controller, a seconds simulation at Reynolds number, kinematic viscosity,7 / and controller start time s was started. For first two seconds no control input was applied to the system. For this time interval the vortex shedding behind the cylinder is visible and the U-V velocities of the flow field at t=.s can be seen in Figure.

5 Finally at the end of the simulation (t=s) the U-V velocities can be seen in the Figure Figure. U-V Velocities at t=.s Figure. U-V Velocities at t=s After seconds the controller starts to effect the system by injecting air from the upper and lower side of the cylinder and the flow field at t=s at the can be observed in Figure 9. It is clear from the figure that the swirling vortices propagating streamwise have been suppressed for the most part. This achievement has come with the expense of the production of a channel of fluid flowing straight past the cylinder, seen in the middle-lower portion of the figure. One should however not criticize this outcome since this portion is not part of the measurement zone so neither the model, nor the controller has knowledge of this flow zone.... In addition to the U-V velocities, the mean vorticity values of the flow field at the specified times were also plotted to show the controller s effect of the flow regime (Figures -) Figure 9. U-V Velocities at t=s At t=9 s, the reduction in the vorticity value starts to be observable as can be seen in Figure Figure. Vorticity at t=.s. - Figure. U-V Velocities at t=9s

6 Figure 3. Vorticity at t=s Figure. Vorticity at t=9s - Figure. Vorticity at t=s Mean vorticity magnitude (/s) Time (s) Figure. Measured Mean Vorticity Magnitude Values Against Time. Conclusion and Future Works In this paper dynamical modeling of the flow past a circular cylinder and controlling the vorticity behind it based on this model was studied. Through numerical simulations it was confirmed that the model represents the flow accurately and the controller system reduces the vorticity magnitude values within the desired area of the flow field by using velocity actuation through selected points over the cylinder. Future research directions include the application of the flow control strategies considered to different geometries such as square cylinders and airfoils. Acknowledgements This work is supported by the Scientific & Technological Research Council of Turkey (TUBITAK) under project 9E33 and by the European Commission (EC) under project PIRG--GA-393. Also the mean vorticity magnitude value behind the circular cylinder was recorded during simulation and it can be examined in Figure. Figures - also confirm the reduction of the swirling vortices in the desired measurement region. References [] U. O. Ünal & Ö. Göre, Girdap yaratıcıların dairesel silindir etrafındaki akışa etkisi, İTÜ Dergisi/D Mühendislik, (), 9, -7. [] B. Protas, Linear feedback stabilization of laminar vortex shedding based on a point vortex model, Physics of Fluids, (),. [3] B. Çuhadaroğlu, Geçirgen yüzeylerinden üfleme yapılan kare kesitli silindir etrafındaki akışta ısı 7

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